Mathematical Sciences: Fourier Analysis and Partial Differential Equations

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Three major areas will be addressed during the course of this mathematical research; geometry, analysis and economic modeling. The Yamabe problem of Riemannian geometry asks if a manifold with a given Riemannian metric is conformally equivalent to another with constant scalar curvature if the new metric is obtained from the old by multiplication by a positive function. Work will be done on the analogous problem for Cauchy-Riemann structures. Here one seeks to choose a Levi form with constant pseudohermitian scalar curvature. This amounts to solving an explicit partial differential equation on the manifold. Results on compact strictly pseudoconvex orientable manifolds have been extensive for odd dimensions. Work will proceed on the remaining cases. Efforts will also be made to extend fundamental (P, q1) - estimates between power integrals of a function and the image of the function under the heat operator. Results over the past two decades, while sharp, have rested on the assumption that the two exponents are in duality. Such results all fall within the general definition of Sobolev inequalities. The renewed interest in such comparisons is related to work on uniqueness properties of solutions of partial differential equations. In addition to seeking new Sobolev inequalities work will also be done in expanding their applicability to more general differential operators. Recent interest in economics derives from a result of Frobenius about differential equations which, in modern terminology, states when a one-form has an integrating factor. In microeconomics one considers one-forms of differences between differentials of income and demand functions of income and prices (multiplied by differentials of prices). These forms distinguish whether or not a tangent vector is pointing in the direction of improvement. The Frobenius result is the statement that a utility function exists - it is generally regarded as a basic axiom. There is evidence to suggest that the axiom fails for collections of consumers. The present project will focus on characterizing conditions when approximate integrability can be expected.